# Coordinate Geometry - Conic Section Test 7

Total Questions:50 Total Time: 75 Min

Remaining:

## Questions 1 of 50

Question:The centre of the conic represented by the equation $$2{x^2} - 72xy + 23{y^2} - 4x - 28y - 48 = 0$$ is

$$\left( {\frac{{11}}{{15}},\;\frac{2}{{25}}} \right)$$

$$\left( {\frac{2}{{25}},\;\frac{{11}}{{25}}} \right)$$

$$\left( {\frac{{11}}{{15}},\; - \frac{2}{{25}}} \right)$$

$$\left( { - \frac{{11}}{{25}},\; - \frac{2}{{25}}} \right)$$

## Questions 2 of 50

Question:The centre of $$14{x^2} - 4xy + 11{y^2} - 44x - 58y + 71 = 0$$

(2, 3)

(2, – 3)

(– 2, 3)

(– 2, – 3)

## Questions 3 of 50

Question:A parabola passing through the point $$( - 4,\; - 2)$$ has its vertex at the origin and y-axis as its axis. The latus rectum of the parabola is

6

8

10

12

## Questions 4 of 50

Question:The focus of the parabola $${x^2} = - 16y$$ is

(4, 0)

(0, 4)

(–4, 0)

(0, –4)

## Questions 5 of 50

Question:The equation of the latus rectum of the parabola $${x^2} + 4x + 2y = 0$$ is

$$2y + 3 = 0$$

$$3y = 2$$

$$2y = 3$$

$$3y + 2 = 0$$

## Questions 6 of 50

Question:Vertex of the parabola $$9{x^2} - 6x + 36y + 9 = 0$$ is

$$(1/3,\; - 2/9)$$

$$( - 1/3,\; - 1/2)$$

$$( - 1/3,\;1/2)$$

$$(1/3,\;1/2)$$

## Questions 7 of 50

Question:The length of the latus rectum of the parabola $$9{x^2} - 6x + 36y + 19 = 0$$

36

9

6

4

## Questions 8 of 50

Question:The axis of the parabola $$9{y^2} - 16x - 12y - 57 = 0$$ is

$$3y = 2$$

$$x + 3y = 3$$$$2x = 3$$

$$y = 3$$

None of these

## Questions 9 of 50

Question:The length of the latus rectum of the parabola $${x^2} - 4x - 8y + 12 = 0$$ is

4

6

8

10

## Questions 10 of 50

Question:The focus of the parabola $$y = 2{x^2} + x$$ is

(0, 0)

$$\left( {\frac{1}{2},\;\frac{1}{4}} \right)$$

$$\left( { - \frac{1}{4},\;0} \right)$$

$$\left( { - \frac{1}{4},\;\frac{1}{8}} \right)$$

## Questions 11 of 50

Question:The point of contact of the tangent $$18x - 6y + 1 = 0$$ to the parabola $${y^2} = 2x$$is

$$\left( {\frac{{ - 1}}{{18}},\;\frac{{ - 1}}{3}} \right)$$

$$\left( {\frac{{ - 1}}{{18}},\;\frac{1}{3}} \right)$$

$$\left( {\frac{1}{{18}},\;\frac{{ - 1}}{3}} \right)$$

$$\left( {\frac{1}{{18}},\;\frac{1}{3}} \right)$$

## Questions 12 of 50

Question:The equation of the common tangent of the parabolas $${x^2} = 108y$$ and $${y^2} = 32x$$, is

$$2x + 3y = 36$$

$$2x + 3y + 36 = 0$$

$$3x + 2y = 36$$

$$3x + 2y + 36 = 0$$

## Questions 13 of 50

Question:The angle between the tangents drawn at the end points of the latus rectum of parabola $${y^2} = 4ax$$, is

$$\frac{\pi }{3}$$

$$\frac{{2\pi }}{3}$$

$$\frac{\pi }{4}$$

$$\frac{\pi }{2}$$

## Questions 14 of 50

Question:The line $$y = mx + c$$ touches the parabola $${x^2} = 4ay$$, if

$$c = - am$$

$$c = - a/m$$

$$c = - a{m^2}$$

$$c = a/{m^2}$$

## Questions 15 of 50

Question:If $$lx + my + n = 0$$ is tangent to the parabola $${x^2} = y$$, then condition of tangency is

$${l^2} = 2mn$$

$$l = 4{m^2}{n^2}$$

$${m^2} = 4\ln$$

$${l^2} = 4mn$$

## Questions 16 of 50

Question:The equation of the tangent to the parabola $${y^2} = 9x$$ which goes through the point (4, 10), is

$$x + 4y + 1 = 0$$

$$9x + 4y + 4 = 0$$

$$x - 4y + 36 = 0$$

$$9x - 4y + 4 = 0$$

3 and 4 are correct

## Questions 17 of 50

Question:The point on the parabola $${y^2} = 8x$$ at which the normal is inclined at 60o to the x-axis has the co-ordinates

$$(6,\; - 4\sqrt 3 )$$

$$(6,\;4\sqrt 3 )$$

$$( - 6,\; - 4\sqrt 3 )$$

$$( - 6,\;4\sqrt 3 )$$

## Questions 18 of 50

Question:The slope of the normal at the point $$(a{t^2},\;2at)$$ of the parabola $${y^2} = 4ax$$, is

$$\frac{1}{t}$$

t

t

$$- \frac{1}{t}$$

## Questions 19 of 50

Question:Equation of any normal to the parabola $${y^2} = 4a(x - a)$$ is

$$y = mx - 2am - a{m^3}$$

$$y = m\,(x + a) - 2am - a{m^3}$$

$$y = m\,(x - a) + \frac{a}{m}$$

$$y = m\,(x - a) - 2am - a{m^3}$$

## Questions 20 of 50

Question:Tangents drawn at the ends of any focal chord of a parabola $${y^2} = 4ax$$ intersect in the line

$$y - a = 0$$

$$y + a = 0$$

$$x - a = 0$$

$$x + a = 0$$

## Questions 21 of 50

Question:For the above problem, the area of triangle formed by chord of contact and the tangents is given by

8

$$8\sqrt 3$$

$$8\sqrt 2$$

None of these

## Questions 22 of 50

Question:The point on parabola $$2y = {x^2}$$, which is nearest to the point (0, 3) is

($$\pm$$ 4, 8)

$$( \pm 1,\,1/2)$$

($$\pm$$ 2, 2)

None of these

## Questions 23 of 50

Question:The equation of the ellipse whose centre is at origin and which passes through the points (-3, 1) and (2, -2) is

$$5{x^2} + 3{y^2} = 32$$

$$3{x^2} + 5{y^2} = 32$$

$$5{x^2} - 3{y^2} = 32$$

$$3{x^2} + 5{y^2} + 32 = 0$$

## Questions 24 of 50

Question:If the eccentricity of an ellipse be 5/8 and the distance between its foci be 10, then its latus rectum is

39/4

12

15

37/2

## Questions 25 of 50

Question:The lengths of major and minor axis of an ellipse are 10 and 8 respectively and its major axis along the y-axis. The equation of the ellipse referred to its centre as origin is

$$\frac{{{x^2}}}{{25}} + \frac{{{y^2}}}{{16}} = 1$$

$$\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{{25}} = 1$$

$$\frac{{{x^2}}}{{100}} + \frac{{{y^2}}}{{64}} = 1$$

$$\frac{{{x^2}}}{{64}} + \frac{{{y^2}}}{{100}} = 1$$

## Questions 26 of 50

Question:If the centre, one of the foci and semi-major axis of an ellipse be (0, 0), (0, 3) and 5 then its equation is

$$\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{{25}} = 1$$

$$\frac{{{x^2}}}{{25}} + \frac{{{y^2}}}{{16}} = 1$$

$$\frac{{{x^2}}}{9} + \frac{{{y^2}}}{{25}} = 1$$

None of these

## Questions 27 of 50

Question:The foci of $$16{x^2} + 25{y^2} = 400$$ are

$$( \pm 3,\;0)$$

$$(0,\; \pm 3)$$

$$(3,\; - 3)$$

$$( - 3,\;3)$$

## Questions 28 of 50

Question:Eccentricity of the ellipse $$9{x^2} + 25{y^2} = 225$$ is

$$\frac{3}{5}$$

$$\frac{4}{5}$$

$$\frac{9}{{25}}$$

$$\frac{{\sqrt {34} }}{5}$$

## Questions 29 of 50

Question:The equation of ellipse whose distance between the foci is equal to 8 and distance between the directrix is 18, is

$$5{x^2} - 9{y^2} = 180$$

$$9{x^2} + 5{y^2} = 180$$

$${x^2} + 9{y^2} = 180$$

$$5{x^2} + 9{y^2} = 180$$

## Questions 30 of 50

Question:In an ellipse the distance between its foci is 6 and its minor axis is 8. Then its eccentricity is

$$\frac{4}{5}$$

$$\frac{1}{{\sqrt {52} }}$$

$$\frac{3}{5}$$

$$25{x^2} + 144{y^2} = 900$$

## Questions 31 of 50

Question:The co-ordinates of the foci of the ellipse $$3{x^2} + 4{y^2} - 12x - 8y + 4 = 0$$ are

(1, 2), (3, 4)

(1, 4), (3, 1)

(1, 1), (3, 1)

(2, 3), (5, 4)

## Questions 32 of 50

Question:The eccentricity of the curve represented by the equation $${x^2} + 2{y^2} - 2x + 3y + 2 = 0$$ is

0

1/2

$$1/\sqrt 2$$

$$\sqrt 2$$

## Questions 33 of 50

Question:If any tangent to the ellipse $$\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$$cuts off intercepts of length h and k on the axes, then $$\frac{{{a^2}}}{{{h^2}}} + \frac{{{b^2}}}{{{k^2}}} =$$

0

1

1

None of these

## Questions 34 of 50

Question:If the line $$y = mx + c$$touches the ellipse $$\frac{{{x^2}}}{{{b^2}}} + \frac{{{y^2}}}{{{a^2}}} = 1$$, then $$c =$$

$$\pm \sqrt {{b^2}{m^2} + {a^2}}$$

$$\pm \sqrt {{a^2}{m^2} + {b^2}}$$

$$\pm \sqrt {{b^2}{m^2} - {a^2}}$$

$$\pm \sqrt {{a^2}{m^2} - {b^2}}$$

## Questions 35 of 50

Question:The value of $$\lambda$$, for which the line $$2x - \frac{8}{3}\lambda y = - 3$$ is a normal to the conic $${x^2} + \frac{{{y^2}}}{4} = 1$$ is

$$\frac{{\sqrt 3 }}{2}$$

$$\frac{1}{2}$$

$$- \frac{{\sqrt 3 }}{2}$$

$$\frac{3}{8}$$

## Questions 36 of 50

Question:The pole of the straight line $$x + 4y = 4$$ with respect to ellipse $${x^2} + 4{y^2} = 4$$ is

(1, 4)

(1, 1)

(4, 1)

(4, 4)

## Questions 37 of 50

Question:The equation of the hyperbola whose conjugate axis is 5 and the distance between the foci is 13, is

$$25{x^2} - 144{y^2} = 900$$

$$144{x^2} - 25{y^2} = 900$$

$$144{x^2} + 25{y^2} = 900$$

$$25{x^2} + 144{y^2} = 900$$

## Questions 38 of 50

Question:The length of the transverse axis of a hyperbola is 7 and it passes through the point (5, -2). The equation of the hyperbola is

$$\frac{4}{{49}}{x^2} - \frac{{196}}{{51}}{y^2} = 1$$

$$\frac{{49}}{4}{x^2} - \frac{{51}}{{196}}{y^2} = 1$$

$$\frac{4}{{49}}{x^2} - \frac{{51}}{{196}}{y^2} = 1$$

None of these

## Questions 39 of 50

Question:The locus of the centre of a circle, which touches externally the given two circles, is

Circle

Parabola

Hyperbola

Ellipse

## Questions 40 of 50

Question:The foci of the hyperbola $$2{x^2} - 3{y^2} = 5$$, is

$$\left( { \pm \frac{5}{{\sqrt 6 }},\;0} \right)$$

$$\left( { \pm \frac{5}{6},\;0} \right)$$

$$\left( { \pm \frac{{\sqrt 5 }}{6},\;0} \right)$$

None of these

## Questions 41 of 50

Question:Centre of hyperbola $$9{x^2} - 16{y^2} + 18x + 32y - 151 = 0$$ is

(1, –1)

(–1, 1)

(–1, –1)

(1, 1)

## Questions 42 of 50

Question:The equation of the hyperbola whose foci are (6, 4) and (-4, 4) and eccentricity 2 is given by

$$12{x^2} - 4{y^2} - 24x + 32y - 127 = 0$$

$$12{x^2} + 4{y^2} + 24x - 32y - 127 = 0$$

$$12{x^2} - 4{y^2} - 24x - 32y + 127 = 0$$

$$12{x^2} - 4{y^2} + 24x + 32y + 127 = 0$$

## Questions 43 of 50

Question:The equation of the tangent to the hyperbola $$2{x^2} - 3{y^2} = 6$$which is parallel to the line $$y = 3x + 4$$, is

$$y = 3x + 5$$

$$y = 3x - 5$$

$$y = 3x + 5$$ and $$y = 3x - 5$$

None of these

## Questions 44 of 50

Question:The locus of the point of intersection of any two perpendicular tangents to the hyperbola is a circle which is called the director circle of the hyperbola, then the equation of this circle is

$${x^2} + {y^2} = {a^2} + {b^2}$$

$${x^2} + {y^2} = {a^2} - {b^2}$$

$${x^2} + {y^2} = 2ab$$

None of these

## Questions 45 of 50

Question:Let E be the ellipse $$\frac{{{x^2}}}{9} + \frac{{{y^2}}}{4} = 1$$ and C be the circle $${x^2} + {y^2} = 9$$. Let P and Q be the points (1, 2) and (2, 1) respectively. Then

Q lies inside C but outside E

Q lies outside both C and E

P lies inside both C and E

P lies inside C but outside E

## Questions 46 of 50

Question:The length of the chord of the parabola $${y^2} = 4ax$$ which passes through the vertex and makes an angle $$\theta$$ with the axis of the parabola, is

$$4a\cos \theta \,{\rm{cose}}{{\rm{c}}^2}\,\theta$$

$$4a{\cos ^2}\theta \,{\rm{cosec}}\,\theta$$

$$a\cos \theta \,{\rm{cose}}{{\rm{c}}^2}\,\theta$$

$$a{\cos ^2}\theta \,{\rm{cosec}}\,\theta$$

## Questions 47 of 50

Question:The locus of the point of intersection of lines $$(x + y)t = a$$ and $$x - y = at$$, where t is the parameter, is

A circle

An ellipse

A rectangular hyperbola

None of these

## Questions 48 of 50

Question:The equation of the hyperbola referred to its axes as axes of coordinate and whose distance between the foci is 16 and eccentricity is $$\sqrt 2$$, is

$${x^2} - {y^2} = 16$$

$${x^2} - {y^2} = 32$$

$${x^2} - 2{y^2} = 16$$

$${y^2} - {x^2} = 16$$

## Questions 49 of 50

Question:The eccentricity of the hyperbola $$\frac{{{x^2}}}{{16}} - \frac{{{y^2}}}{{25}} = 1$$ is

3/4

3/5

$$\sqrt {41} /4$$

$$\sqrt {41/5}$$

## Questions 50 of 50

Question:The equation to the hyperbola having its eccentricity 2 and the distance between its foci is 8

$$\frac{{{x^2}}}{{12}} - \frac{{{y^2}}}{4} = 1$$
$$\frac{{{x^2}}}{4} - \frac{{{y^2}}}{{12}} = 1$$
$$\frac{{{x^2}}}{8} - \frac{{{y^2}}}{2} = 1$$
$$\frac{{{x^2}}}{{16}} - \frac{{{y^2}}}{9} = 1$$